I'm not sure I get it... An exponential rate of increase "reaches infinity" (a former math teacher of mine would scold me for using this terminology, as infinity isn't a place that can be reached) only after an infinite amount of time.

The singularity is the point where the AI robots start building smarter AI robots - once they get going, the increase is exponential, and this is the singularity - you end up with super-intelligent robots, with intelligence far beyond what we as humans are capable of.